effects of three-dimensional geometry and radial electric ... · effects of three-dimensional...
TRANSCRIPT
Effects of Three-Dimensional Geometry and Radial
Electric Field on ITG Turbulence and Zonal Flows
GROKINETICS IN LABORATORY AND ASTROPHYSICAL PLASMASWorkshop “Kinetic-scale turbulence in laboratory and space plasmas: emprical constraints,
fundamental concepts and unsolved problems” Issac Newton Institute for Mathematical Sciences, Cambridge, July 23, 2010
H. Sugama
National Institute for Fusion Science,Graduate University of Advanced Studies
Toki 509-5292, Japan
in collaboration with T.-H. Watanabe M. Nunami
OUTLINE
Introduction
Linear ITG Mode Analysis for High-Ti LHD plasmas
Zonal Flows and ITG Turbulence in Helical Systems
Effects of Equilibrium Electric Field Er on Zonal Flows in Helical Systems
Momentum Balance and Radial Electric Field in Quasisymmetric Systems with Stellarator Symmetry
Summary
Introduction
Tokamak
Helical System
B = B0 [ 1 − εt cos θ − εh cos (Lθ −Mζ) ]
B
Classification ofparticle orbits
passing
passing
trapped
trapped
!
vdr b
B = B0 (1 − εt cos θ )
vdr radial drift of helically-trapped particles
toroidally-trapped particles
Watanabe et al. NF2007
Eigenfunction of linear ITGmode electrostatic potential
Sugama & Watanabe PoP2006
Zonal-flow response(GAM, residual ZF)
Helical geometry influences ITG mode and zonal flow.
Gyrokinetic Equations (for ITG Turbulence)
!
"
"t+ v
||ˆ b # $ + vd # $ %µ ˆ b # $&( ) "
"v||
'
( )
*
+ , -f +
c
B0
.,-f{ } = v/ % vd % v||ˆ b ( ) # e$.
TiFM + C -f( )
!
v" = #cT
i
eLnB
0
1+$i
mv2
2Ti
#3
2
%
& '
(
) *
+
, -
.
/ 0 ̂ y , µ =
v12
22
!
J0
k"v" #( )$f d3v% & 1&'
0k"
2( )[ ]e(
Ti
=e
Te
( & (( ) , k"2 = kx + ˆ s zky( )
2
+ ky
2
Diamagnetic drift
Quasineutrality condition & Adiabatic electron assumption
Ion gyrokinetic equation for
Ion polarization
!
k"#i $1, k"#e <<1
δ f(x, v||, µ, t)
!
vd" #Gyrocenter drift
Mirror force
!
"µ b # $%( ) & /&v||
Effects of magnetic geometry
Linear ITG Mode Analysis forHigh-Ti LHD plasmas
Results from Linear ITGMode Analyses by GKV-X
(See Poster by M. Nunami)
Zonal Flows and ITG Turbulencein Helical Systems
For low collisionality, better confinement is observed in theinward-shifted magnetic configurations, where lower neoclassicalripple transport but more unfavorable magnetic curvature drivingpressure-gradient instabilities are anticipated.
H. Yamada et al. (PPCF2001)
Scenario:Neoclassical optimization contributesto reduction of anomalous transportby enhancing the zonal-flow level.
10-1
100
10-1 100 101
Rax=3.6m
Rax=3.75m
Rax=3.9m
!Eexp/!E
ISS95
Reactor
condition
"*b
Anomalous transport is alsoimproved in the inwardshifted configuration.
Results from LHD experiments
Inward-shifted
Standard
Outward-shifted
!
KL (t) =1" (2 /# )1/ 2 (2$H )
1/ 2{1" gi1(t,%}
1+G + E(t) n0k&2'ti
2( )
Collisionless Time Evolution of Zonal Flows in Helical Systems
Response of the zonal-flow potential to a given initial potential
Response function = GAM component + Residual component
GAM response functionLong-time response function
[Sugama & Watanabe, PRL (2005), Phys.Plasmas (2006)]
k ρi < 1
E(t) represents effects of shielding of potential due to helical-ripple-trapped particles.
!
"k(t) = K(t) "
k(0)
!
K(t) = KGAM
(t)[1"KL(0)]+ K
L(t)
!
K(t)"KL(t), K
GAM(t)" 0 as t" +#
!
K(t = 0) =1
!
KGAM
(t) = cos("G)exp(# | $ | t)
(L=2, M=10)
B = B0 [ 1 − εt cos θ − εh cos (Lθ −Mζ) ]
!
E(t) =2
"n0(2#H )
1/ 2{1$ gi1(t,%)} $
3
2k&2'ti
2(2#H )
1/ 2{1$ gi1(t,%)}
(
) *
+Ti
Te(2#H )
1/ 2{1$ ge1(t,%)}
+
, -
Linear time evolution of zonal-flow potential
Smaller χi and larger zonal flows arefound in the saturated turbulent state forthe inward-shifted configuration than forthe standard one !
Turbulent thermal diffusivityand squared zonal-flow potential
Larger residual zonal flow is foundfor the inward-shifted case.
!"#$
"
"#$
"#%
"#&
"#'
(
" (" $" )" %" *" &" +" '"
Standard ( kr ! = 0.25)
Inward ( kr ! = 0.28)
,"-./0123,"
-."12
/000.Ln034
/0i001
inward-shiftedinward-shifted configurationconfiguration standard configurationstandard configuration
Results from GKV simulation ( flux tube, Er = 0 )
Watanabe, Sugama & Ferrando, PRL(2008)Sugama, Watanabe & Ferrando, PFR(2009)
The GKV turbulence simulations were carriedout by the Earth Simulator (JAMSTEC).
ITGturbulence
Potential contours obtained from six copies of flux tube
Effects of Equilibrium Electric Field Er
on Zonal Flows in Helical Systems
In helical systemsEr is given from ambipolar condition of radial particle fluxes.Er reduces neoclassical ripple transport.
How does Er influence zonal flows and anomalous transport?
Effects of Er on gyrokinetic equation and zonal flows
Gyrokinetic equation for
!
k" = kr#r
!
"
" t+ v||
ˆ b # $ + ik r # vd %µ ˆ b # $&( )"
"v||
+'E
"
"(
)
* +
,
- . /f = %ik r # vd
e 0(x + 1)
TiFM
!
"E
= #c E
r
r0B0
angular velocity due to ExB drift
!
" = # $% /q
gyrophase average ofzonal-flow potential
field line label
In helical systems, α -dependence appears in and .
!
µ ˆ b " #$( )
!
kr" v
d
Therefore, even if the zonal-flow potential φ is independent of α , δ f comes to depend on α .
Thus, ωE influences δ f and accordingly φ through quasineutrality condition.
Classification of particle orbits in the presence of Er
θ
κ2 trapping parameter
π−π 0
κ2=1
passinghelicallytrapped(closed)
ExB drift
ΔE
helicallytrapped(unclosed)
toroidallytrapped
ExB drift transition
pointsΔE
radial displacement ofhelically-trapped particle
!
"E~ r
0
vdr
vE#B
!
" 2 =1# $B
0[1#%
T(&) #%
H(&)]
2$B0%H(&)
trapping parameter
Cary et al., PF (1988)Wakatani (1998)
!
" =1
2mv
2 µ
passing
helically trapped (closed)
helically trapped(unclosed)
toroidally trapped
Solution of gyrokinetic equation to describe long-timeevolution of zonal flows [Sugama & Watanabe, PoP(2009)]
!
" fk (t) = #e
T$k (t)FM 1# e
# i kr % r e#i kr & r e
i kr & r J0(kr%r )orbit
[ ]
+ e#i kr % r e# i kr & r ei kr & r " fk
(g )(0) + FM Sk (t)dt
0
t
'[ ]orbit
Perturbed particle distribution function
Polarization(classical & neoclassical)
Initial condition &Turbulence source
!
Lorbit
= Ldl
(dl /dt)"
dl
(dl /dt)"
Average along the orbit
For particles which show transitions
!
dl
(dl /dt)" = (1# P
t)
dl
v||$ 2>1v ||>0
" +dl
v||$ 2>1v ||<0
" + Pt
d%
&E$ 2<1
"
toroidallytrapped
helicallytrapped
Pt transition probability
1-Pt
helicallytrapped
toroidallytrapped
ExB drift transition
points
!
"rL
#rL
gyro motiondrift motion
1-Pt
Pt 1
!
Pt "4 2
#
c Er
v B0
$
% &
'
( ) R0q
r0
$
% &
'
( ) *H( )
+1/ 2 ,*H ,-
, *H +*T( ) ,-
.
/ 0
1
2 3 P+
Zonal-flow generation can be enhanced whenGht and Gh decreases with neoclassical optimization (which reduces radial drift velocity vdr )and when poloidal Mach number increases
with increasing Er and using heavier ions.
Long-time zonal-flow response to theinitial condition and turbulence source
!
e
Ti"k (t) =
n0#1
d3v 1+ i kr $ i r # $ i r
orbit( )[ ] % f i k(g )(0) + FM Sik (t)dt0
t
&[ ]&
kr'ti( )21+Gp +Gt + Mp
#2(Gh t +Gh )(1+ Te /Ti)[ ]
For radial wavenumbers krρti < 1 (ITG turbulence) and krΔE < 1, the zonal-flowpotential is derived from the quasineutrality condition as [Sugama & Watanabe, PoP(2009)]
!
Mp " (cEr B0) (rvt i Rq)
Geometrical factors G’s represents shielding effects of neoclassical polarization due toparticles motions in different orbits.
Gp : passingGt : toroidally-trapped
Ght : helicallly-trapped (unclosed orbit)
Gh : toroidally-trapped (closed orbit)
!
G " (population)
# $r%( )
2
No transitions occur.
Response to the initial condition
!
"k (t) ="k (0)
1+Gp +Gt + Mp
#2(Gh t +Gh )(1+ Te /Ti)
Assume the initial distribution to have Maxwellian dependenceThen, we obtain
For the single-helicity configuration
!
B = B0[1"#t cos$ "#h cos(L$ "M% )] (#h: independent of$)
!
" fi k(g )(0) = [" nk
(g )(0) /n
0]FM
" nk(g )(0) /n
0= (kr#ti)
2e$(0) /Ti
(no turbulence source)
!
"k (t) ="k (0)
1+Gp +Gt + (15# /4)Mp
$2q2(2%h )
1/ 2(1+ Te /Ti)!
Gh t = 0, Gh = (15" /4)q2(2#h )1/ 2
This corresponds to the case considered by previous works.Mynick & Boozer, PoP(2007)Sugama, Watanabe & Ferrando, PFR(2008)
The residual zonal-flow potential as a function of kr ρtifor Mp = 0 and Mp = 0. 3
Theoretical results are derived
by assuming kr ρti << 1 .
Different kr ρti dependences
for Mp = 0 and Mp = 0. 3 aretheoretically predicted andconfirmed by simulation.
Inward-shited configuration
[To be published in CPP]
Momentum Balance and Radial Electric Field inQuasisymmetric Systems with Stellarator Symmetry
Basic Boltzmann Kinetic Equation for description ofCollisional and Turbulent Transport
Equilibrium magnetic field
Boltzmann kinetic equation
Ensemble-averaged kinetic equation
Classical, Neoclassical, and Anomalous Transportof Particles and Heat [Sugama et al. PoP1996]
The gyrophase (ξ) -average part and the oscillating part of an aribtrary
function F is defined by and respectively.
The ensemble-averaged kinetic equation is divided as
Particle flux
Heat flux
Second order part of in δ ~ ρ / L
Momentum Balance
density particle flux
pressure tensor
friction force
turbulent electromagnetic force
Momentum Balance in the direction tangential to the flux surface
!
"
" tnam
ac1ua# +
(SEM)#
c2
$
% &
'
( ) + c2 ua* +
(SEM)*
c2
$
% &
'
( )
+ , -
. / 0 a
1
= 21
V '
"
" sV ' 3s 4 P
a
a
1 2TEM
$
% &
'
( ) 4 c1
" x
"#+ c
2
" x
"5
$
% &
'
( )
6
7 8 8
9
: ; ;
+1
c2c
1<'+c
2='( ) e
anaua
s
a
1
(c1, c2 : constants)
(s, θ, ζ) : Hamada coordinates
The surface-averaged radial current
Quasisymmetry[Boozer(1983), Nuhrenberg(1988), Helander&Simakov (2008)]
quasi-axi-symmetry (c1, c2) = (0, 1) quasi-poloidal-symmetry (c1, c2) = (1, 0)
The ambipolarity
is satisfied automatically up to O(δ).
The O(δ) viscosity component in the quasisymmetrydirection vanishes :
Stellarator Symmetry
Magnetic field strength
Metric tensor components
Magnetic field components
Parity Transformation associated with Stellarator Symmetry
Expansion in η ~ δ ~ ρ / L
Parity operator is defined by
In the presence of stellarator symmetry, Boltzmann andMaxwell equations are invariant under parity transformation
!
ea"#$1
ea
(Put in Boltzmann and Maxwell eqs.)
Momentum Transport Fluxes in Stellarator Symmetric Systems
Parity of solutions
When j is even, the O(δ j ) part of radial transport fluxesof poloidal and toroidal momentum vanish.
Momentum Balance in Quasisymmetric Systemswith Stellarator Symmetry
In quasisymmetric systems with stellarator symmetry,the momentum transport fluxes vanish up to O(δ 2 ), andthe ambipolarity is automatically satisfied up to O(δ 2 ).
!
= "1
V '
#
# sV ' $s % P
a
(3)
a
& "TEM
(3)'
( )
*
+ , % c1
# x
#-+ c
2
# x
#.
'
( )
*
+ ,
/
0 1 1
2
3 4 4
The momentum balance equation determing Es is of O(δ 3 ) :
Summary Fluctuations observed in a high Ti LHD plasma are considered as ITG modes predicted from linear calculation by GKV-X.
Zonal-flow response theory and simulation show that zonal flow generation and turbulence regulation are enhanced when the radial displacements of helical-ripple-trapped particles are reduced either by neoclassical optimization of the helical geometry lowering the radial drift velocity or by strengthening the radial electric field Er to boost the poloidal rotation.
The Er effects appear through the poloidal Mach number Mp. For the same magnitude of Er, higher zonal-flow response is obtained by using ions with heavier mass (favorable deviation from gyro-Bohm scaling).
The momentum balance equation determining Er in quasisymmetric helical system with stellarator symmetry is shown to be of O(δ 3 ) by using a novel parity operator.